SemiDefinite Programming in AMPL?

[MAURO VIEGAS DA SILVA]

How implementation Semidefinite relaxations in ampl?

[e.d.an...@mosek.com]

According to my knowledge it is not possible.

[If it is possible to handle SDPs in AMPL we at MOSEK would update our AMPL link to handle so MOSEK could be used to solve them.]

[Robert Fourer]

There isn't currently any way to specify a semidefiniteness constraint in AMPL.

If semidefinite relaxations are being used for quadratic programming, then it would be possible to formulate a quadratic program in AMPL and have the solver interface convert to the semidefinite formulation -- this makes sense if you view semidefinite relaxation as a solution method rather than a modeling technique. I am not aware of any current interface that actually does this, however.

Bob Fourer
am...@googlegroups.com

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[MAURO VIEGAS DA SILVA]

The used of mosek, is possible modeling in ampl? 

[MAURO VIEGAS DA SILVA]

My problem used conic in ampl, but the problem very is complex, many paper recommend semidefinite ,  my questions is possible modelar this format? For exemple, modeling in ampl the solver xpress fico?

[e.d.an...@mosek.com]

Yes, but not for conic problems.

[Robert Fourer]

There is no straightforward way in AMPL to specify that a "matrix" of variables must be positive semidefinite.  In principle it would be possible to create a user-defined suffix, say .psd, such that all variables with the same .psd value would be treated by the solver interface as a matrix to be conveyed to the solver with a positive semi-definite restriction; but this has not been done in any solver interfaces that I know of, and since there isn't a matrix data type in AMPL, it would depend upon some assumptions about the ordering of variables.  A different approach using a general nonlinear solver is described in the paper On Formulating Semidefinite Programming Problems as Smooth Convex Nonlinear Optimization Problems by Vanderbei and Benson; that relies on a user-defined function (C source is available) and also makes some assumptions about variable ordering.

Bob Fourer
am...@googlegroups.com

[e.d.an...@mosek.com]

I would like to mention one caveat about the suffixes. If you assign a suffix to variable but that variable is NOT used anywhere in the model, then it is removed from the model when the NL file is written. [Ref. personal communication with David Gay.] Such variables can occur in semidefinite optimization as conic quadratic optimization models.  

I know this because I tried to use suffixes for conic quadratic models, but had to give up that approach for the reason mentioned above.

I do not think there are any good suggestion for extend traditional modeling languages like to handle conic problems e.g. SDPs.

[Robert Fourer]

I agree that extensions based on existing modeling language facilities (such as suffixes or user-defined functions) have too many weaknesses. Conceivably new facilities the specifically express semi-definiteness restrictions could be built into modeling language syntax, so that the language processor would have the information needed to provide a correct problem instance to the SDP solver. But it's not trivial to incorporate a matrix concept into a language designed for scalar formulations.

Bob Fourer
am...@googlegroups.com

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